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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

6 Issues per year


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Convergence rates and source conditions for Tikhonov regularization with sparsity constraints

D. A. Lorenz1

1 Zentrum für Technomathematik, Fachbereich 3, Universität Bremen, PO Box 330440, 28334 Bremen, Germany. Email:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 16, Issue 5, Pages 463–478, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/JIIP.2008.025, September 2008

Publication History

Received:
2008-01-25
Revised:
2008-04-23
Published Online:
2008-09-12

Abstract

This paper addresses the regularization by sparsity constraints by means of weighted ℓp penalties for 0 ≤ p ≤ 2. For 1 ≤ p ≤ 2 special attention is payed to convergence rates in norm and to source conditions. As main results it is proven that one gets a convergence rate of in the 2-norm for 1 < p ≤ 2 and in the 1-norm for p = 1 as soon as the unknown solution is sparse. The case p = 1 needs a special technique where not only Bregman distances but also a so-called Bregman-Taylor distance has to be employed.

For p < 1 only preliminary results are shown. These results indicate that, different from p ≥ 1, the regularizing properties depend on the interplay of the operator and the basis of sparsity. A counterexample for p = 0 shows that regularization need not to happen.

Key words.: Sparsity constraint; ill-posed problems; Tikhonov regularization

Citing Articles

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[2]
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[3]
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Journal of Optimization Theory and Applications, 2015, Volume 165, Number 1, Page 78
[4]
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[5]
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[6]
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Stephan W Anzengruber, Bernd Hofmann, and Ronny Ramlau
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Stephan W. Anzengruber, Bernd Hofmann, and Peter Mathé
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[10]
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[11]
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[12]
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Inverse Problems, 2013, Volume 29, Number 6, Page 065018
[13]
Martin Burger, Jens Flemming, and Bernd Hofmann
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[14]
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[15]
Bernd Hofmann and Peter Mathé
Inverse Problems, 2012, Volume 28, Number 10, Page 104006
[16]
Markus Grasmair
Journal of Mathematical Analysis and Applications, 2010, Volume 365, Number 1, Page 19
[17]
V Kolehmainen, M Lassas, K Niinimäki, and S Siltanen
Inverse Problems, 2012, Volume 28, Number 2, Page 025005
[18]
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[19]
Dinh Nho Hào and Tran Nhan Tam Quyen
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[20]
Klaus Frick, Dirk A. Lorenz, and Elena Resmerita
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[21]
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[22]
Stephan W Anzengruber and Ronny Ramlau
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[23]
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[24]
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[28]
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[29]
D. A. Lorenz and D. Trede
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[30]
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Communications on Pure and Applied Mathematics, 2011, Volume 64, Number 2, Page 161
[31]
[32]
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Inverse Problems, 2010, Volume 26, Number 11, Page 115009
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[34]
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[35]
Kristian Bredies and Dirk A Lorenz
Inverse Problems, 2009, Volume 25, Number 8, Page 085011
[36]
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